---
tag: 563ANQ3MY7
kind: concept
name: Wright omega function
msc: ["33E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Wright_omega_function
same_as:
  wikidata: Q8038235
  wikipedia: Wright omega function
---

## Summary

The Wright omega function $\omega$ is a single-valued function built from the branches of the [[Lambert W function|FW2JKAN14B]].
It is $\omega(z) = W_{K(z)}(e^z)$, where the unwinding number $K(z)$ picks the branch.
It solves the equation $\omega + \ln \omega = z$ for all $z$ off two horizontal half-lines.
On the real line it is the unique real solution of this equation.
It is convenient in computer algebra because it avoids the choice of a branch.

## Explanations

### L1 human

The Lambert W function has many branches, and it is easy to pick the wrong one.
The Wright omega function packages the right branch for the input $e^z$ into one function of $z$.
On the real line it is simple: $\omega(x)$ is the positive number $y$ with $y + \ln y = x$.
For example, $\omega(1) = 1$ because $1 + \ln 1 = 1$.